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Imagine driving along a road at night: if there's a missing bridge or the road suddenly jumps ten feet into the air, your ride comes to a sudden halt.
In geometry, you might have heard that a continuous curve is one you can draw without lifting your pencil, but calculus requires an exact mathematical test.
In 1817, mathematician Bernard Bolzano realized that intuitive terms like "unbroken" caused major errors in mathematical proofs, leading him to formulate the first precise definition of continuity.
How do we translate that physical idea of an unbroken road into a rigorous three-step test?

The Three-Part Definition of Continuity

A function f(x)f(x) is continuous at a point x=cx = c if and only if it satisfies three strict conditions: f(c)f(c) is defined, the limit lim⁡x→cf(x)\lim_{x \to c} f(x) exists, and lim⁡x→cf(x)=f(c)\lim_{x \to c} f(x) = f(c).
If even a single one of these three conditions fails, the function is discontinuous at that point.
Let's test these three conditions on a piecewise function across a boundary point.
Consider the function f(x)={2x+1if x<37if x=3x2−2if x>3f(x) = \begin{cases} 2x + 1 & \text{if } x < 3 \\ 7 & \text{if } x = 3 \\ x^2 - 2 & \text{if } x > 3 \end{cases}. Let's determine if ff is continuous at x=3x = 3.
Step 1: Check if f(3)f(3) is defined. By definition, f(3)=7f(3) = 7, which is a real number.
Step 2: Check if the limit exists. The left-hand limit is lim⁡x→3−(2x+1)=7\lim_{x \to 3^-}(2x + 1) = 7, and the right-hand limit is lim⁡x→3+(x2−2)=7\lim_{x \to 3^+}(x^2 - 2) = 7. Because both one-sided limits match, lim⁡x→3f(x)=7\lim_{x \to 3} f(x) = 7.
Step 3: Compare the limit to the function value. Since lim⁡x→3f(x)=7=f(3)\lim_{x \to 3} f(x) = 7 = f(3), all three conditions pass, so f(x)f(x) is continuous at x=3x = 3.